How would someone go about making a "Tag" function?

You should probably just check the distance between players instead of having each of their parts have a touched event and if you want it to be similar to jailbreak.

local runService = game:GetService("RunService")
local contextService = game:GetService("ContextActionService")

local ply = game.Players.LocalPlayer
local char = ply.Character
local ui = ply.PlayerGui.TagUi  --Assuming this is a ScreenGui

ply.CharacterAdded:connect(function(c)
    char = c
end)

function tweenUi()
    --Put Code to tween your Ui and tag progress 
end

function searchTarget() --Check if player is in range
    local dis = 3
    local tar = nil

    for _,v in pairs(game.Players:GetChildren()) do
        if v.Character and v.Character ~= char then
            local tDis = (v.Character.PrimaryPart.Position-char.PrimaryPart.Position).magnitude
            if tDis < dis then
                dis = tDis
                tar = v
            end
        end
    end
    if target then
        ui.Enabled = true
        target = tar
        contextService:BindAction("tagTween",tweenUi,false,Enum.KeyCode.E)  --Binding tag function to key
    else
        ui.Enabled = false
        contextService:UnbindAction("tagTween")
    end
end

runService:BindToRenderStep("findTarget",0,searchTarget)    --More functionality than while loop